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# 核磁共振成像学习笔记——从FID信号到K空间

[left{ begin{aligned} S_c(t)=M_{xy}e^{-t/T_2}cos({omega}t)\ S_s(t)=M_{xy}e^{-t/T_2}sin({omega}t) end{aligned} right. ]

[left{ begin{aligned} cos({omega}t)=frac{e^{i{omega}t}+e^{-{i{omega}t}}}{2}\ sin({omega}t)=frac{e^{i{omega}t}-e^{-{i{omega}t}}}{2i} end{aligned} right. ]

[ S(t)=S_c(t)+iS_s(t) ]

[ S(t)=M_{xy}*e^{-t/T_2}*e^{-i{omega}t} ]

[ S(t)=M_{xy}*e^{-i{omega}t} ]

[ omega={gamma}B ]

[ S(t)=iint{rho(x,y)e^{-iphi(x,y,t)}}dxdy ]

[ phi(x,y,t)={gamma}int_{0}^{t}{[G_x(t^{prime})x+G_y(t)y]}dt^{prime} ]

[left{ begin{aligned} k_x=2piint_{0}^{t}{G_x(t^{prime})}dt^{prime}\ k_y=2piint_{0}^{t}{G_y(t)}dt^{prime} end{aligned} right. ]

[ S(t)=iint{rho(x,y)e^{-i2pi[k_xx+k_yy]}}dxdy ]

[ S(k_x,k_y)=iint{rho(x,y)e^{-i2pi[k_xx+k_yy]}}dxdy ]

(S_c(t))信号为例，其组成的是(S(t))中的实部，在没有进行空间编码前

[left{ begin{aligned} k_x=2piint_{0}^{t}{G_x(t^{prime})}dt^{prime}\ k_y=2piint_{0}^{t}{G_y(t)}dt^{prime} end{aligned} right. ]

[1]MRI From Picture to Proton
[2]MRI, The Basics
[3]MRI磁振影像學 盧家鋒
[4]MRI原理-信号 - lemon lelieven的文章 - 知乎 https://zhuanlan.zhihu.com/p/137255997
[5]【磁共振的K空间】 https://www.bilibili.com/video/BV1ch411e7Yc/?share_source=copy_web&vd_source=0e8c3fe50c67df43ceeb30f63e36eb0d